Remark 2.1 We stress that equation (REF ) is not a classical Wasserstein-2 gradient flow in the probability space. The classical gradient flow looks for one probability function of \(\theta \)  in the pseudo-time \(s\)  [1]}. Instead, we now have \(\rho (\theta ,t)\) , where \(\rho \)  is a probability density of \(\theta \)  for every \(t\) , and their flows are all coupled through the definition of \(E\) . Note that this feature emerges exactly because we have infinitely many layers \(L\rightarrow \infty \) , reformulated using a continuous variable \(t\) . When there are only 2 layers, the dependence on \(t\)  can be dropped, and the problem reduces to the classical gradient flow [2]}.
A core obstacle to be overcome in this paper is the need for new technicalities to handle the non-standard gradient flow.

